A note on rational homology vanishing theorem for hypersurfaces in aspherical manifolds

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In this note, Gromov’s reduction [No metrics with positive scalar curvatures on aspherical 5-manifolds, https://arxiv.org/abs/2009.05332, 2020], from the aspherical conjecture to the generalized filling radius conjecture, is generalized to the smooth Q \mathbb Q -homology vanishing conjecture in the case of hypersurface. In particular, we can show that any continuous map from a closed 4 4 -manifold admitting positive scalar curvature to an aspherical 5 5 -manifold induces zero map between H 4 ( ⋅ , Q ) H_4(\cdot ,\mathbb Q) . As a corollary, we obtain the following aspherical splitting theorem: if a complete orientable aspherical Riemannian 5 5 -manifold has non-negative scalar curvature and two ends, then it splits into the Riemannian product of a closed flat manifold and the real line.